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Principles and brief history ofCavity QED
S.Haroche,1st specialized Solvay Lecture,
June 4 2010
From the Bohr-Einstein photonbox thought experiment…
…to the super-high Q cavities oftoday’s real experiments….
…exploring the quantum dynamicsof atoms and photons in a confined
space has progressed a lot…
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…and its «realization»by Gamov…
Bohr’s draft of a box storing andreleasing a photon to test quantumlaws…
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1
Early History of Cavity QED:controlling spontaneous emission
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Early History: Tailoring spontaneous emissionin a confined space
Spontaneous processes are random. Only their rate can be predicted and,in the case of photon emission, estimated by classical arguments based on
Maxwell’s equations
The spontaneous emission rate of an excited state depends on the atom’sstate, but also on the structure of the surrounding vacuum, which
determines the density of modes into which photons are emitted: an atomwithin boundaries does not radiate as in free space.
Similar effects in beta decay: a neutron lives longer in a nucleus than infree space!
Spontaneous emissionenhancement predicted byE.Purcell in 1945…
…and the possibility to inhibitspontaneous emission in atoms
suggested by D.Kleppner in 1981
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Emissioninhibited
Emissionenhanced
When gap is increased to l = l/2, modedensity jumps and undergoesresonances for larger l values
100-1000
or larger
2l/!
Antennae radiating near reflecting surfaces
Dispersive effects:cavity Lamb shifts,
Casimir effect l/!
Free-spacemode
density
Modedensity in
cavity (peakincreaseswith Q)
"c/"
In close-spaced gap, field
modes with polarization \\ to
mirrors are suppressed
Dipole \\
to mirror
and
image
cancell
Dipole #
to mirror
and
image
add up
"c/"
When mirrors are curved,focusing effects enhancethe resonances, leading to
huge emissionenhancement factors
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First demonstration of Purcell effect on atoms
Rydberg atoms prepared instate 23S in a cavity (V=70mm3)
resonant with transition 23S-
22P (!=340 GHz).
Ionization signal in a ramped electric field applied
to the atoms after they leave the cavity.
The 22P state ionizes in a larger field, thus at a
later time than 23S.
Ionization of 22P
Ionization of 23S
Signals corresponding to anaverage of N atoms crossing
together the cavity: N = 3.5, 2
and 1.3 for traces a,b,c
respectively. Cavity on
resonance (solid line) or off-
resonance (dashed line).Enhancement factor:
! = "at
C/"(23S#22P ) = 530
P.Goy, J-M.Raimond, M.Gross et S.Haroche, PRL 50, 1903 (1983)
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Inhibiting the spontaneous emission of circularRydberg atoms
Atom prepared in circularRydberg state n=22
(orbit parallel to metal plates)
Ionisation
detector
Atomic transmission versus "/2L: " is
swept by Stark effect, L being kept
constant. The sharp signal increase for"/2L=1 demonstrates the inhibition of s.e.
of the Rydberg atom which survives
longer in its initial state.
microwave transition
Inhibited transition n=22 # n’=21 at " = 0.45mm
! / 2L
R.G.Hulet, E.S.Hilfer et D.Kleppner,PRL 55, 2137 (1985).
Many enhancement and inhibition experiments in microwave,infrared and optical part of spectrum realized since these
pionneering studies…
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Collective emission in cavity: from Purcell to Dicke
Atoms located at equivalent nodal positions incavity are symetrically coupled to field: they
evolve during emission in a subspace invariant byatomic permutation. There is no way to know
which atom has emitted when a photon is lost…
Strong correlations with entanglement spontaneously build up betweenatoms, making collective dipole larger than when atoms radiate
independently
Superradiance rate proportional to number N of atoms
Due to this correlation, the spontaneous emission occurs faster than forsingle atom: this is Dicke superradiance
Purcell factor: ~ number ofimages in cavity wall collectively
emitting with one atom
Number of atomsradiating collectively
together
A double enhancement effect: !C(N ) ="N!
0
e,e ! "S =1
2e,g + ge( ) ! g,gTwo atoms:
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Observation of Dicke superradiance in a cavity
Sample of N=3200 Sodium atoms prepared
in Rydberg state 29S, emitting collectively in
a cavity resonant with 29S-28P transition at!= 162 GHz. The single atom spontaneous
emission rate in free space on this transitionis $0 =43s-1. Purcell factor: % = $at
C / $0 ~ 70.
The atom-cavity coupling is switched-off after variable
time by applying an electric field in cavity (Stark
effect). For each interaction time t , we measure thenumber of atoms in states 29S and 28P after cavity
exit. From an ensemble of 900 realizations of
experiment, we reconstruct the histograms of the
number Ne of excited atoms as a function of t (in unitsof tD ~ %N/$0 = 460 ns).
Agreement between experimental histograms andtheory (solid lines in black)
J-M Raimond, P.Goy, M.Gross, C.Fabre et S.Haroche, Phys.Rev.Lett. 49, 1924 (1982)
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2.
The strong coupling regime of CQED intime-domain:
Rydberg-atom microwave experiments
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From Purcell to Rabi:the strong coupling regime of Cavity QED
Spontaneous emission in acontinuum of cavity modes of
width &c = '/Q imparts toatomic excited state a width $c
inversely proportional to &c.
&c$c
(
!c="2
#c
<< #c
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From Purcell to Rabi:
the strong coupling regime of Cavity QED
As cavity Q factor increases, thecavity spectral width &c='/Q
decreases and the rate ofemission $c shoots up. The
perturbative treatment of the
Purcell effect breaks down when
these two widths become equal.
&c$c
(
!c="2
#c
$ #c
! =da .E0
h= da
"
2h#0Vc
$ %c ="
Q
Atomicdipole
Vacuumfluctuations in
Cavity
Strong coupling
regime:
large dipole, small cavityvolume and very large Q
factor
VacuumRabi
frequency
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e
gThe spin:
2-level atom
From Purcell to Rabi:the strong coupling regime of Cavity QED is
a story about a spin and a spring
0
1
2
The spring:
Cavity mode
H =h!
eg
2e e " g g#$ %&+ h!a†a " i
h'
2a e g " a† g e#$ %&
(Jaynes Cummings Hamitonian)
e,0 !"! cos#t
2
$
%&
'
() e,0 + sin
#t
2
$
%&
'
() g,1
Vacuum Rabi oscillation:(reversible spontaneous emission)
e,n !"! cos# n +1t
2
$
%&
'
() e,n + sin
# n +1t
2
$
%&
'
() g,n +1
Rabi oscillation sped-up in n
photons (stimulated emission)
(
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n
p(n)
0 1 2 3
Rabi oscillation in vacuum or in small coherent field:direct test of photon graininess
Pe(t) = p(n)cos
2 ! n +1t
2
"
#$
%
&'
n
( ; p(n) = e)n n
n
n!
n = 0 (nth = 0.06)
n = 0.40 (±0.02)
n = 0.85 (±0.04)
n = 1.77 (±0.15)
Pe(t) signal Fourier transform Inferred p(n)
Brune et al,PRL,76,1800,1996.
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First strong coupling experiment in CQED:the micromaser (1985)
Herbert Walther
1935-2006Meschede et al, PRL 54, 551 (1985)
Rydberg atoms cross one at a time a
high Q cavity and build up a many-
photon field in it by cumulative Rabi
oscillations:
the ultimate maser-laser
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The ideal micromaser: a quantum machine todeliver photons in a box
!n = 0
!n = +1
j = 0 j =1
Probabilities given by Rabi:
Pj(n) = cos
2 ! n +1t + j"
2
#
$%
&
'(
Photon numberconverges to n0
Simulations:n undergoes
staircase-likeevolution, varying
randomlybetweendifferent
realizations
Solid line:ensemble average
If trapping
condition
fulfilled, all
trajectories
converge to n0
(here n0=10) Photon nber histograms at increasing times
! n0+1t = 2p"
# Pj=1(n0 ) = 0
Trapping states
l
t = l / v
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The two-photon micromaser:cavity tuned at half-frequency of transition between same parity levels
!n = 0
Emits photons by pairsSingle photon emission towards intermediate level is inhibited by CQED
M.Brune et al, PRL 59, 1899 (1987)
!n = +2
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J.McKeever et al, Nature, 425, 268 (2003).
Lasing of a single
atom trapped in a
cavity
(Caltech group)
Microlasers in optical CQED
The optical version of the micromaser:
field builds up from « kicks » produced
by atoms crossing one by one the cavity
K.An et al, PRL, 73, 3375 (1994).
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3.
The strong coupling regime of atomic-CQED in optical experiments
H.J.Kimble (Caltech), G.Rempe (Garching),T.Esslinger (ETH-Zurich)
Chapman (Georgia Tech), Vuletic (MIT),Orozco (Maryland), Blatt (Insbruck),Meschede (Bonn), Lange (Sussex)…
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Cavity QED in optical domain:the atom-cavity «!molecule!»
The transmission spectrum of thecavity is split into two componentswhen cavity contains a single atom
(from atomic beam or dropped froma MOT).
Fourier transform of time-dependent Rabi oscillation
Thompson et al, PRL, 68, 1132 (1992)
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Single atom detection by cavity field
transmission
Depending on laser frequency, a single atom
transit across cavity is signaled by a dip or a
peak.
A 100% efficient atom detector which can count
one by one atoms in the cavity
a
a
b b
J.McKeever et al, PRL 93, 143601 (2004)
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Using CQED as single atom counter to studyatom-laser statistics
similarity with optical laser (Glauber theory)
Second order atom
correlation (BEC atom-laser)
Histogram of number of
atoms in time-bin (Poisson)
Second order atom
correlation for thermal
atom beam
A.Öttl et al, PRL, 95, 090404 (2005)
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Conversely, the atom modifies the cavityfrequency, which changes its response to the
field. From an analysis of this fieldtransmission, the position of the atom inside
the cavity can be obtained in real time:an atom-cavity microscope tracking atoms.
Feed back procedures to improve thetrapping have been implemented.
C.J.Hood et al, Science, 287, 1457
(2000) .T.Fischer et al, PRL 88, 163002
(2002)
Trapping force of a single photon in acavity
The atom-cavity dressed energies are atomic
position dependent. Their spatialderivative corresponds to a light-force
exerted on average by one photon!This force can attract atom inside the
cavity.
Garching group (similar films by Caltech group)
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The CQED photon pistol:releasing photons one by one on demand
A Raman process on a single atom converts
triggering pulses into photons escaping one
by one from the cavityi
frepumping
Experiment performed
with flying atoms, then
with trapped atoms in
cavity
Kuhn et al, PRL, 89, 067901 (2002) J.McKeeveret al, Science, 303, 1992 (2004)M. Hijlkema et al,arXiv:quant-ph/0702034 (2007)
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The escaping light isanalysed after beam
splitting by correlation.Coincidence rate measured
as a function of delaybetween the two output
channels.
Missing central peak isevidence of single photon
emission(a photon cannot be
«!split!»)
Analyzing the photon pistol output
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Single atom-non linear optics:the photon blockade effect in optical CQED
Once a photon is resonant with theatom-cavity system, a second photon isoff-resonant: only one photon at a timecan be transmitted by cavity atresonance!
The transmitted light isantibunched
Birnbaum et al, Nature 436, 87 (2005).
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4. Entanglement experiments in microwave
CQED
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Two essential ingredients
Circular Rydberg atoms
Large circular orbit
Strong coupling to microwaves
Long radiative lifetimes (30ms)
Level tunability by Stark effect
Easy state selective detection
Quasi two-level systems
n = 51
n = 50
Superconducting microwave cavity
Gaussian field mode with 6mm waist
Large field per photon
Long photon life time improved by ring around mirrors (1ms)
Easy tunability
Possibility to prepare Fock or coherent states with controlled
mean photon number
e
g
The
spring
The
spin
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Oven Circular Rydberg
state preparation
Cavity
Microwave source
(coherent state)
Auxiliary microwave
(atom manipulation)
State selective
detection
Artist’s view of the Paris microwave CQED
set-up (2001-2005 version)
Raimond, Brune and Haroche RMP, 73, 565 (2001)
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The route to circular states: a 53 photon adiabatic process
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Rubidium level scheme with transitions
implied in the selective depumping and
repumping of one velocity class in the
F=3 hyperfine state
In green, velocity distribution before
pumping, in red velocity distribution of
atoms pumped in F=3, before being excited
in circular Rydberg state
Controlling the atom-cavity interaction time: atomic velocity
selection by optical pumping
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0 30 60 900.0
0.2
0.4
0.6
0.8
Pe(t
)
time ( ? s)
51 (level e)
50 (level g)
51.1 GHz
Hagley et al, PRL 79, 1 (97)
Initial state
|e,0>
$ / 2 pulse
Creates atom-cavity
entanglement
|e,0> % |e,0> + |g,1>
µ
Useful Rabi pulses
( quantum knitting)
e,0 ! cos"t
2
#$%
&'(e,0 + sin
"t
2
#$%
&'(g,1
EPR pairs in CQED
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0 30 60 900.0
0.2
0.4
0.6
0.8
Pe(t
)
time ( ? s)
51 (level e)
50 (level g)
51.1 GHz
Maître et al, PRL 79, 769 (97)
|e,0> % |g,1>
|g,1>% |e,0>
|g,0> % |g,0>
(|e> +|g>)|0> % |g> (|1> +|0>)
µ
e,0 ! cos"t
2
#$%
&'(e,0 + sin
"t
2
#$%
&'(g,1
$ pulse maps
atomic state on
field and back
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0 30 60 900.0
0.2
0.4
0.6
0.8
Pe(t
)
time ( ? s)
51 (level e)
50 (level g)
51.1 GHz
Nogues et al, Nature, 400, 239 (1999); Rauschenbeutel et al, PRL, 83, 5166 (1999)
2$ pulse: phase gate and
quantum logic operations
|e,0> % - |e,0>
|g,1> % - |g,1>
|g,0> % |g,0>
µ
e,0 ! cos"t
2
#$%
&'(e,0 + sin
"t
2
#$%
&'(g,1
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V(t)
e1g2
Entangled atom-atom pair mediated by real photon exchange
Electric field F(t) used to
tune atoms #1 and #2 in
resonance with C for a
determined time t realizing
)/2 or ) Rabi pulse
conditions
Hagley et al, P.R.L. 79,1 (1997)
+
+
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Direct entanglementof two atoms via
virtual photonexchange:
a cavity-assistedcontrolled collision(after S.B.Zeng and G.C.Guo, PRL 85,
2392 (2000)).
Relatively insensitive to cavity Q and thermal photons
! = 10"6#eg
1
$1
+1
$2
%
&'
(
)*
Two modes:1/*#1/* 1 +1/*2
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A thought experiment about complementarity
– Microscopic slit: set in motion when deflecting particle.Which path information and no fringes
– Macroscopic slit: impervious to interfering particle.No which path information and fringes
– Wave and particle are complementary aspects of the quantum object.
Particle/slit
entanglement
Einstein-Bohr
discussionat Solvay
1927
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A “modern” version of Bohr’s proposal
with a Mach-Zehnder interferometer
_
_
D
•Interference between two well-separated paths.
• Getting a which-path
information?
&
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A “modern” version of Bohr’s proposal:
Mach-Zehnder with a moving beam-splitter
• Massive beam splitter: negligible motion, no which- path information, fringes
• Microscopic beam splitter: which path information and no fringes
_
_
D&
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Complementarity and entanglement
• A more general analyzis of Bohr’s experiment
– Initial beam-splitter state
– Final state for path b
– Particle/beam-splitter state
– Particle/beam-splitter entanglement
– (an EPR pair if states orthogonal)
– Final fringes signal
• Small mass, large kick
NO FRINGES
• Large mass, small kick
FRINGES
0
!
0a b
!" = " + "
0a b
!" "
0 0! =
0 1! =
_
O
B1
B2
M
M'
a
b
P
D
_&
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Complementarity and decoherence
Entanglement with another system destroys interference
• explicit detector (beam-splitter/ external)
• uncontrolled measurement by the environment (decoherence)
_
_
D
Complementarity, decoherence and entanglement intimately linked
&
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A more realistic system: Ramsey interferometry• Two resonant $/2 classical pulses on an atomic transition e/g
_B2
B1
M
M'
a
b
D
Which path information?
Atom emits one photon in R1 or R2
Ordinary macroscopic fields
(heavy beam-splitter)
Field state not appreciably affected. No "which path" information
FRINGES
Mesoscopic Ramsey field
(light beam-splitter)
Addition of one photon changes the field. "which path" info
NO FRINGES
R1 R2
0 10 20 30 40 50 60
0.0
0.2
0.4
0.6
0.8
1.0
Pg
Fréquence relative (kHz)
&
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An object at the quantum/classical boundary
Coherent field in a cavity
• State produced by a classicalsource coupled for finite time tothe cavity mode: field defined by
complex amplitude '
• A picture in phase space(Fresnel plane)
From quantum to classical
• Vacuum or
small field:• Large quantum fluctuations. A field at the
single-photon level is a quantum object
• Large field
• Small quantum fluctuations. A field withmore than 10 photons is almost aclassical object.
a
!'
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Bohr’s experiment with a Ramsey interferometerStore one Ramsey field in a high Q cavity
Initial cavity state
– Intermediate atom-cavity state
• Ramsey fringes contrast
– Large field
• FRINGES
– Small field
NO FRINGE
D
S
R1R2
e
g
C
__
!
( )1, ,
2e ge g! !" = +
e g! !
e g! ! !" "
0 , 1e g
! != =
Atom-cavity interaction timeTuned for $/2 pulse
Possible even if C empty
From
quantum to
classicalclassical
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Quantum/classical limit for an interferometer
Fringes contrast versus photon number N in first Ramsey field
Fringes vanish for quantum
field
photon number plays
the role of the beam-
splitter's "mass"
An illustration of the (N()
uncertainty relation :
• Ramsey fringes reveal
field pulses phase
correlations.
• Small quantum field: large
phase uncertainty and low
fringe contrast
Not a trivial blurring of the
fringes by a classical noise:
atom/cavity entanglement
can be erasedNature, 411, 166 (2001)
0 2 4 6 8 10 12 14 160.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Fringes c
ontr
ast
N
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An elementary quantum eraser
• Another thought experiment
_
_
D
Two interactions with the same beamsplitter assembly erase the which path information
and restore the interference fringes
&
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Ramsey “quantum eraser”
• A second interaction with the mode erases the which path info
Ramsey fringes without fields !
– Quantum interference fringes without external field
– A good tool for quantum manipulations
_,0e
( )1
,0 ,12e g+
,0e
10 12 14 16 18 20 22 24
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Pe
|g,1>
Atom found in g: one photon in C
whatever the path:no info and fringes
&
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A conditional quantum
eraser: new perspective
on EPR